Question: $ C = \left[\begin{array}{rrr}-1 & 3 & 0 \\ 4 & 0 & 5\end{array}\right]$ $ D = \left[\begin{array}{rr}-2 & 5 \\ 2 & 2 \\ 3 & -1\end{array}\right]$ What is $ C D$ ?
Answer: Because $ C$ has dimensions $(2\times3)$ and $ D$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ C D = \left[\begin{array}{rrr}{-1} & {3} & {0} \\ {4} & {0} & {5}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{5} \\ {2} & \color{#DF0030}{2} \\ {3} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{3}\cdot{2}+{0}\cdot{3} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{3}\cdot{2}+{0}\cdot{3} & ? \\ {4}\cdot{-2}+{0}\cdot{2}+{5}\cdot{3} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{3}\cdot{2}+{0}\cdot{3} & {-1}\cdot\color{#DF0030}{5}+{3}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{-1} \\ {4}\cdot{-2}+{0}\cdot{2}+{5}\cdot{3} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{-2}+{3}\cdot{2}+{0}\cdot{3} & {-1}\cdot\color{#DF0030}{5}+{3}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{-1} \\ {4}\cdot{-2}+{0}\cdot{2}+{5}\cdot{3} & {4}\cdot\color{#DF0030}{5}+{0}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}8 & 1 \\ 7 & 15\end{array}\right] $